Some applications of stochastic averaging method for quasi Hamiltonian systems in physics

2009 ◽  
Vol 52 (8) ◽  
pp. 1213-1222 ◽  
Author(s):  
MaoLin Deng ◽  
WeiQiu Zhu
Keyword(s):  
Hamiltonian Systems ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method

STOCHASTIC HOPF BIFURCATION OF QUASI-INTEGRABLE HAMILTONIAN SYSTEMS WITH FRACTIONAL DERIVATIVE DAMPING

2012 ◽  
Vol 22 (04) ◽  
pp. 1250083 ◽  
Author(s):  
F. HU ◽  
W. Q. ZHU ◽  
L. C. CHEN
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Hamiltonian Systems ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Bifurcation Parameter ◽  
Integrable Hamiltonian Systems ◽  
Fractional Derivative Damping ◽  
Stochastic Hopf Bifurcation

The stochastic Hopf bifurcation of multi-degree-of-freedom (MDOF) quasi-integrable Hamiltonian systems with fractional derivative damping is investigated. First, the averaged Itô stochastic differential equations for n motion integrals are obtained by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, an expression for the average bifurcation parameter of the averaged system is obtained and a criterion for determining the stochastic Hopf bifurcation of the system by using the average bifurcation parameter is proposed. An example is given to illustrate the proposed procedure in detail and the numerical results show the effect of fractional derivative order on the stochastic Hopf bifurcation.

Random Response of Wake-Oscillator Excited by Fluctuating Wind

2021 ◽  
pp. 1-33
Author(s):  
Mao Lin Deng ◽  
Genjin Mu ◽  
Weiqiu Zhu
Keyword(s):  
Hamiltonian Systems ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stochastic Averaging Method ◽  
Random Response ◽  
Integrable Hamiltonian Systems ◽  
Wake Oscillator ◽  
Fluctuating Wind ◽  
Oscillator Models

Abstract Many wake-oscillator models applied to study vortex-induced vibration (VIV) are assumed to be excited by ideal wind that is assumed to be uniform flow with constant velocity. While in the field of wind engineering, the real wind generally is described to be composed of mean wind and fluctuating wind. The wake-oscillator excited by fluctuating wind should be treated as a randomly excited and dissipated multi-degree of freedom (DOF) nonlinear system. The involved studies are very difficult and so far there are no exact solutions available. The present paper aims to carry out some study works on the stochastic dynamics of VIV. The stochastic averaging method of quasi integrable Hamiltonian systems under wideband random excitation is applied to study the Hartlen-Currie wake-oscillator model and its modified model excited by fluctuating wind. The probability and statistics of the random response of wake-oscillator in resonant or lock-in case and in non-resonant case are analytically obtained, and the theoretical results are confirmed by using numerical simulation of original system. Finally, it is pointed out that the stochastic averaging method of quasi integrable Hamiltonian systems under wideband random excitation can also be applied to other wake-oscillator models, such as Skop-Griffin model and Krenk-Nielsen model excited by fluctuating wind.

Recent Developments and Applications of the Stochastic Averaging Method in Random Vibration

1996 ◽  
Vol 49 (10S) ◽  
pp. S72-S80 ◽  
Author(s):  
W. Q. Zhu
Keyword(s):  
Hamiltonian Systems ◽  
Random Vibration ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Degree Of Freedom ◽  
New Development ◽  
Recent Developments

Comprehensive surveys on the stochastic averaging method in random vibration until the late 1980s were given by Roberts and Spanos (1986) and Zhu (1988). The present paper reviews the recent developments and applications of the stochastic averaging method in random vibration since then. A major new development of the stochastic averaging method in recent years is the generalization of the method to multi-degree-of-freedom, quasi-Hamiltonian systems.

Stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Lincong Chen ◽  
Fang Hu ◽  
Weiqiu Zhu
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Hamiltonian Systems ◽  
Stochastic Dynamics ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Integrable Hamiltonian Systems ◽  
Fractional Optimal Control ◽  
Fractional Derivative Damping

AbstractIn the present survey, some progress in the stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping is reviewed. First, the stochastic averaging method for quasi integrable Hamiltonian systems with fractional derivative damping under various random excitations is briefly introduced. Then, the stochastic stability, stochastic bifurcation, first passage time and reliability, and stochastic fractional optimal control of the systems studied by using the stochastic averaging method are summarized. The focus is placed on the effects of fractional derivative order on the dynamics and control of the systems. Finally, some possible extensions are pointed out.

An improved energy envelope stochastic averaging method and its application to a nonlinear oscillator

2016 ◽  
Vol 23 (1) ◽  
pp. 119-130 ◽  
Author(s):  
Yaping Zhao
Keyword(s):  
Steady State ◽  
Probability Density Function ◽  
Probability Density ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
System Response ◽  
Mean Square ◽  
Fpk Equation ◽  
The Mean

An improved stochastic averaging method of the energy envelope is proposed, whose application sphere is extensive and whose implementation is convenient. An oscillating system with both nonlinear damping and stiffness is taken into account. Its averaged Fokker-Planck-Kolmogorov (FPK) equation in respect of the transition probability density function of the energy envelope is deduced by virtue of the method mentioned above. Under the initial and boundary conditions, the joint probability density function as to the displacement and velocity of the system is worked out in closed form after solving the averaged FPK equation by right of a technique based on the integral transformation. With the aid of the special functions, the transient solutions of the probabilistic characteristics of the system response are further derived analytically, including the probability density functions and the mean square values. A simple approach to generate the ideal white noise is drastically ameliorated in order to produce the stationary wide-band stochastic external excitation for the Monte Carlo simulating investigation of the nonlinear system. Both the theoretical solution and the numerical solution of the probabilistic properties of the system response are obtained, which are extremely coincident with each other. The numerical simulation and the theoretical computation all show that the time factor has a certain influence on the probability characteristics of the response. For example, the probabilistic distribution of the displacement tends to be scattered and the mean square displacement trends toward its steady-state value as time goes by. Of course the transient process to reach the steady-state value will obviously be shorter if the damping of the system is greater.

The Effects of Correlated Noise on Bifurcation in Birhythmicity Driven by Delay

2018 ◽  
Vol 28 (10) ◽  
pp. 1850127 ◽  
Author(s):  
Lijuan Ning ◽  
Zhidan Ma
Keyword(s):  
Averaging Method ◽  
Harmonic Approximation ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Self Control ◽  
Correlated Noise ◽  
State Variables ◽  
Stationary Probability Density Function ◽  
Fpk Equation ◽  
Control Feedback

We consider bifurcation regulations under the effects of correlated noise and delay self-control feedback excitation in a birhythmic model. Firstly, the term of delay self-control feedback is transferred into state variables without delay by harmonic approximation. Secondly, FPK equation and stationary probability density function (SPDF) for amplitude can be theoretically mapped with stochastic averaging method. Thirdly, the intriguing effects on bifurcation regulations in a birhythmic model induced by delay and correlated noise are observed, which suggest the violent dependence of bifurcation in this model on delay and correlated noise. Particularly, the inner limit cycle (LC) is always standing due to noise. Lastly, the validity of analytical results was confirmed by Monte Carlo simulation for the dynamics.

scholarly journals Averaging Method for Neutral Stochastic Delay Differential Equations Driven by Fractional Brownian Motion

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Peiguang Wang ◽  
Yan Xu
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Delay Differential Equations ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Delay Differential

In this paper, we investigate the stochastic averaging method for neutral stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H∈1/2,1. By using the linear operator theory and the pathwise approach, we show that the solutions of neutral stochastic delay differential equations converge to the solutions of the corresponding averaged stochastic delay differential equations. At last, an example is provided to illustrate the applications of the proposed results.

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